Under the assumptions \(N=\frac{d}{\delta }+c\sqrt{d}+o\left(\sqrt{d}\right)\)  and \(k=\left(2-\frac{1}{\delta }\right) d+ b\sqrt{d}+o\left(\sqrt{d}\right)\) , Godland, Kabluchko and Thäle [1]} have determined the limit for \({\mathbb {E}}\,f_k(C_N)/\binom{N}{k}\) . The following theorem improves and supplements their result.
